Moment Methods for Uniform Average Mixing on Strongly Regular Graphs

Abstract:We study continuous-time quantum walks on connected strongly regular graphs that are not complete, observed at a random time drawn from a freely chosen probability law. Uniform average mixing (UAM) asks for a law under which every averaged transition probability equals $1/n$, where $n$ is the number of vertices. On a strongly regular graph this is equivalent to two affine constraints on three cosine moments. We construct a bounded, compactly supported time density for every strongly regular graph with nonintegral eigenvalues. For integral spectra we give an exact finite Toeplitz criterion and its Hankel form. Every averaged mixing matrix of such a graph is realized by at most two observation times. Three elementary inequalities on the moment line, which also give a short proof of Chan's classification of complex Hadamard matrices in the Bose-Mesner algebra, lead to a determination of all strongly regular graphs that admit UAM. Apart from the conference graphs of nonsquare order and the graphs with instantaneous uniform mixing, these are the members of two infinite families of parameter sets and their complements, and for them we give explicit laws with two observation times. The Petersen graph and its complement are the smallest members. A strongly regular graph with UAM admits a bounded time density exactly when it has no instantaneous uniform mixing. We also correct the classification of instantaneous uniform mixing on strongly regular graphs by Godsil, Mullin and Roy. Its sign condition excludes the halved $5$-cube, which mixes uniformly at time $\pi/4$. With the order $4\theta^2$ read literally, its parity condition also excludes the Clebsch graph and includes the parameters $(36,14,4,6)$, for which no time law gives uniform average mixing.

26 pages, 1 table. Includes a correction of Theorem 5.1 and Lemma 5.2 in arXiv:1301.5889

Quantum Physics (quant-ph); Combinatorics (math.CO); Probability (math.PR)

arXiv:2610.10017 [quant-ph]

(or arXiv:2610.10017v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2610.10017 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.